Let/. It :0 ---0 G be any two self maps of a compact connected oriented Lie group G. In this paper, for each positive integer k , we associate an integer with fk,hi . We relate this number with Lefschetz coincidence number. We deduce that for any two differentiable maps f, there exists a positive integer k such that k 5.2+1 , and there is a point x C G such that ft (x) = (x) , where A is the rank of G . Introduction Let G be an n-dimensional com -pact connected Lie group with multip-lication p ( .e 44:0 xG--+G such that p ( x , y) = x.y ) and unit e . Let [G, G] be the set of homotopy classes of maps G G . Given two maps f , f G ---• Jollowing [3], we write f. f 'to denote the map G-.Gdefined by 01.11® =A/WO= fiat® ,sea Given a point g EC and a differ-entiable map F: G G , write GA to denote the tangent space of G at g [4,p.10] , and denote by d x F the linear map rig F :Tx0 T, (x)G induced by F , it is called the differential of Fat g [4,p.22]. Let LA, Rx :0 G be respec-tively the left translation Lx(i)=4..(g,e) , and the right translation Rx(1)./..(gcg). Then there is a natural homomorphism Ad ,the adjoin representation, from G to GL(G•), (the group of nonsingular linear transformations of Qdefined as follows:- Ad(g)= deRe, od,Lx. Note that d xRc, ad.; =d(4,( Lx(e)))0 de; =d.(4, 04)=4(40 Re) = d(4(4, (e)))0 (44, =d ar, o (44, . Since G is connected , the image of Ad belongs to the connected component of G(G)containing the identity,i.e. for each g E 0, detAd(g) > 0 . By Exercise Al • Dr.-Prof.-Department of Mathematics- College of Science- University of Baghdad. •• Dr.-Department of Mathematics- College of Science for Woman- University of Baghdad.
Background: Synthetic hydroxyapatite,(Ca10(PO4)6(OH2) can directly bond to bones without infection and fibrous encapsulation, thus is regarded as bioactive and biocompatible. The aim of the study was the estimation of microarchitecture bone parameters include bone mass (gm/cm2) cortical bone width (mm), thread width (mm), marrow space star volume analysis (V*m) and osteoblast, osteocyte cell number. Materials and methods: Ninety-six (96) commercially pure titanium CpTi) used in this study, (48) implants were coated with HA by dipping coating and (48) implants were used as control. They were inserted in (32) Newzland white rabbits and followed for 2 & 6 weeks. Mechanical torque removal test and histomorphometric analysis of bone microarchit
... Show MoreIn present work an investigation for precise hole drilling via continuous wave (CW) CO2 laser at 150 W maximum output power and wavelength 10.6 μm was achieved with the assistance of computerized numerical controlled (CNC) machine and assist gases. The drilling process was done for thin sheets (0.1 – 0.3 mm) of two types of metals; stainless steel (sst) 321H, steel 33 (st). Changing light and process parameters such as laser power, exposure time and gas pressure was important for getting the optimum results. The obtained results were supported with computational results using the COMSOL 3.5a software code.
To investigate the effects of losartan and enalapril on serum uric acid in hypertensive patients with metabolic syndrome, one hundred and twenty six newly diagnosed mild hypertensive patients, having markers of metabolic syndrome included in the study. The patients were divided into two groups. Group 1 (60 patients) was given losartan (50 mg/ day) and group 2 (66 patients) enalapril (20 mg/ day) for a duration of 2 months. A control group of seventy apparently healthy individuals were included. Metabolic syndrome was diagnosed according to diagnostic criteria of metabolic syndrome related to the American National Cholesterol Education Program-Adult Treatment Panel III. Serum uric acid levels were measured bef
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The Reversed Phase High Performance Liquid Chromatography (RP-HPLC) has been used for the separation of Poly aromatic hydrocarbons(PAHs) by using column Reprosil 100 C 18 which was found to be a suitable one for this purpose. The result showed that using mobile phase of (Acetonitrile-water) Reversed Phase HPLC , flow rate of (1.2 ml/min) , column temperature (30CËš) and wave length of (254nm), give a complete separation with a good resolution . The total separation time was less than 20 min. The result of the study showed that the vegetables of Baghdad city were polluted by poly aromatic hydrocarbons(PAHs) in different places where samples taken because of drainage of the heavy water ,industrial trash and trash of oil colanders. -
... Show MoreIn this work, nanostructure zinc sulfide (ZnS) thin films at temperature of substrate 450 oC and thickness (120) nm have been produced by chemical spray pyrolysis method. The X-Ray Diffraction (XRD) measurements of the film showed that they have a polycrystalline structure and possessed a hexagonal phase with strong crystalline orientation of (103). The grain size was measured using scanning electron microscope (SEM) which was approximately equal to 80 nm. The linear optical measurements showed that ZnS nanostructure has direct energy gap. Nonlinear optical properties experiments were performed using Q-switched 532 nm Nd:YAG laser Z-scan system. The nonlinear refractive index (n2) and nonlinear absorption coefficient (β) estimated for Z
... Show MoreThe galvanic corrosion of the (Cu - Fe), (Cu - Zn) and (Fe - Zn) couples have been investigated in 3.5% NaCl solution, 40ºC, different velocities (Re = 5000, 10000 and 15000) and different area ratio’s of cathode to anode (AR= 0.5,1 and 2), by using commercial metal pipe (cylindrical tube).The Zero Resistance Ammeter has been used to measure the galvanic current (Ig) and galvanic potential (Eg) with time. The galvanic current density increases with increasing velocity (Re) and the area ratio (AR). The galvanic potential (Eg) is shifted to less negative with increasing velocity (Re) and the area ratio (AR). A statistical relations for the galvanic current density and galvanic potential as a function of (Re). and the area ratio had been
... Show MoreIncorporating modern technology into education is becoming imperative. Numerous pharmacy institutions are incorporating virtual reality (VR) technology training into their curricula to enhance educational experience. This review examines the current state, historical evolution, and application of VR programs in pharmacy education and training. The review also provides details about the main challenges and limitations associated with the use of this technology. The VR technology, including virtual laboratories and simulations, significantly improves clinical training and educational outcomes. The utilization of VR in clinical teaching encounters numerous barriers, including ethical concerns and technological constraints, as well as other res
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